1/2+3/4+7/8+15/16+31/32+127/128+255/256

1/2+3/4+7/8+15/16+31/32+127/128+255/256
1/2+3/4+7/8+15/16+31/32+127/128+255/256

1/2+3/4+7/8+15/16+31/32+127/128+255/256
1/2+3/4+7/8+15/16+31/32+127/128+255/256
解,得：
==1/2+(1-1/4)+(1-1/8)+(1-1/16)+(1-1/32)+(1-1/128)+(1-1/256)
==1/2+1-1/4+1-1/8+1-1/16+1-1/32+1-1/128+1-1/256
==6+1/2-(1/4+1/8)-(1/16+1/32)-(1/128+1/256)
==6+1/2-3/8-3/32-3/256
==6+1/8-3/32-3/256
==6+1/32-3/256
==6+5/256
==6又5/256

(1-1/2)+(1-1/4)+……（1-1/256)=8-(1/2+1/4+1/8+……+1/256)
=9-1/256
少了一个1/64

1/2+3/4+7/8+15/16+31/32+127/128+255/256
=1-1/2+1-1/4+1-1/8+1-1/16+1-1/32+1-1/128+1-1/256
=7-(128+64+32+16+8+2+1)/256
=7-251/256
=6又5/256

1/2+3/4+7/8+15/16+31/32+63/64+127/128+255/256
=（1-1/2)+(1-1/4)+(1-1/8)+(1-1/16)+(1-1/32)+(1-1/64)+(1-1/128)+(1-1/256)
=1*8-(1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256)
=8-(1/2+1/4+1/8+1/16+1/3...

全部展开

1/2+3/4+7/8+15/16+31/32+63/64+127/128+255/256
=（1-1/2)+(1-1/4)+(1-1/8)+(1-1/16)+(1-1/32)+(1-1/64)+(1-1/128)+(1-1/256)
=1*8-(1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256)
=8-(1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/256)+1/256
=8-(1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/128)+1/256
=8-(1/2+1/4+1/8+1/16+1/32+1/64+1/64)+1/256
=8-(1/2+1/4+1/8+1/16+1/32+1/32)+1/256
=8-(1/2+1/4+1/8+1/16+1/16)+1/256
=8-(1/2+1/4+1/8+1/8)+1/256
=8-(1/2+1/4+1/4)+1/256
=8-(1/2+1/2)+1/256
=8-1+1/256
=7+1/256
所以，结果为：
七又二百五十六分之一

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